The Branching Plane

The matrix elements of when expanded in a Taylor expansion to first order around the point of conical intersection Rq become  

Using cylindrical polar coordinates x= pcos0, y= psinO, the above Hamiltonian becomes 

the angle a, which relates a diabatic representation to the adiabatic representation, is related to the angle 0 defined from the intersection-adapted coordinates. 

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In Figure 14b we show the potential energy surface for a model elliptic conical intersection46 plotted in the branching plane (xt, x2). Because, as stated earlier, the cone is elliptic in the linear approximation (i.e., the base of the cone is an ellipse rather than a circle), there are two steep sides of the ground state cone surface and two ridges . There are two preferred directions for downhill motion located on the steep sides of the ground state cone surface. A simple... 

Figure 14 Illustration of the general procedure used to locate the initial relaxation direction (IRD) toward the possible decay products, (a) General photochemical relaxation path leading (via conical intersection decay) to three different final structures, (b) Potential energy surface for a model elliptic conical intersection plotted in the branching plane, (c) Corresponding energy profile (as a function of the angle a) along a circular cross section centered on the conical intersection point and with radius d.

Here, a topology-adapted representation [55] was chosen, where (Xi,X2) lift the degeneracy at the intersection and thus span the branching plane [74], These modes are obtained by orthogonalizing the modes (X, Xa) of Eq. (10). The third mode Xg is in turn orthogonal to (Xi, ) and carries information on the intersection space, i.e., the X+ component of Eqs. (9)-(10). Alternative construction schemes are possible in particular, the bilinear coupling terms can be eliminated within the three-mode subspace [54,72]. 

The detailed derivation of iTeff is given in Ref. [53]. Here, two of the six modes (i.e., X and X2) are chosen as topology-adapted modes that span the branching plane for a chosen pair of electronic states (here, states 1 and 2). Each /th-order residual term now also comprises 6 modes,... 

Eq. (A.9) shows how the branching-plane vectors are calculated in practice from Cl difference and transition densities as well as the CSF first-derivative density ... 

At a conical intersection, the branching plane is invariant through any unitary transformation within the two electronic states and any such combination of degenerate states is still a solution. Thus, the precise definition of the two vectors in (A.9) or (A. 11) is not unique and depends on an arbitrary rotation within the space of the Cl coefficients (i.e., between the generalized crude adiabatic states), unless the states have different symmetries (then xi is totally symmetric and X2 breaks the symmetry). 

The elliptic cone model of the potential energy surface at a conical intersection point discussed above is not general enough to give a correct description of the relaxation in realistic molecules. First, more than two possible IRDs may originate from the tip of the cone. Second, the first-order approximation (i.e., elliptic cone) may break down at larger distances, and some IRDs may lie out of the branching plane because the real... 

The pair (x, y) define the branching plane or g-h plane. The remainder of the intersection adapted coordinate system, w , i = l-(Ar " — 2), spans the seam space. These — 2 mutually orthonormal vectors need only be orthogonal to the branching space. It is also convenient to define... 

In Fig. 5 we plot the branching plane vectors (Xi and X2) at the conical intersection of 1. It is apparent that in this molecule Xi and X2 describe either a local torsional deformation of the central segment of the molecule (this second mode can be more rigorously described as coupled... 

Fig. 10. The conical intersection of cyanine dyes is characterized by two electronic states that differ in the position of a positive charge. The intersection mediates a charge-transfer process in the X2 direction of the branching plane dominated by stretching deformations and Z/E isomerization in the Xi direction. The same situation occurs in polyenal protonated Schiff bases (see Scheme 1).

Fig. 22. The S i/5 o conical intersection mediating the photochemical ring-opening of benzopyran together with the Xi and X2 vectors, (right) Schematic representation of the branching plane structure. The values of the relevant structural parameters are given in A and degrees. Data from Ref. 59.

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